Several Classes Of Constrained Matrix Equations And Their Minimum Rank Solutions

The constrained matrix equation problem has been widely used in many fields such as structural design, system identification, automatic control theory, finite elements, vibration theory, and linear optimal control and so on. Solving constrained matrix equations for fixed rank solutions has great significance to perfect the theory of constrained matrix equation.The following problems are considered systematically in this M.S. thesis:ProblemⅠGiven A∈Rp×n, B∈Rn×q, C∈Rp×q, S(?)Rn×n, letDetermine M,m and give the representations of the elements in So and the optimal approximation to a given element.ProblemⅡ(1) Given A, B, D∈Cn×p, letDetermine the representations of the elements in S2 and the optimal approximation to a given element. Where GCSC"X" is a generalized symmetric matrices set.(2) Given A∈Cm×p, B∈Cq×n, D∈Cm×n, letDetermine the maximal and minimal rank of element (X, Y) in S3. When the rank reaches the minimum, give the general expression of the elements (X, Y) in S3.The main achievements are as follows:(1) For ProblemⅠ, when S is central symmetry,anti-central symmetry, symmetry or bisymmetric matrix etc, the maximal or minimal rank of element X in S1 and the representations of the elements in So is obtained by using singular value decomposition,quotient singular value decomposition,matrix partition,matrix structure,Gaussian elimination and relevant theories of rank. The optimal approximate solution to a given element is obtained. (2) For problemⅡ, by mainly using the quotient singular value decomposition, the restricted singular value decomposition (RSVD), the rank inequalities, we obtain the general expression of element X, Y in S2 and the optimal approximation for a given element.The maximal and minimal ranks X, Y in S3 and the representations of the elements in So are obtained.